In Part 1 of the base study (Week 0 to Week 12), participants will be randomized to one of 4 study arms (Arm A: MK-3222 200 mg at Week 0 and Week 4 + matching placebo to etanercept twice weekly; Arm B: MK-3222 100 mg at Week 0 and Week 4 + matching placebo to etanercept twice weekly; Arm C: Matching placebo to MK-3222 at Week 0 and Week 4 + matching placebo to etanercept twice weekly; Arm D: Matching placebo to MK-3222 at Week 0 and Week 4 + etanercept 50 mg twice weekly).